By conditioning on \(\eta \) , we have
\(& {\mathbb {P}}(T_1^{\prime } \le t_1,\ldots ,T_n^{\prime } \le t_n, N(t)=n \mid \eta (u), 0<u \le t ) \nonumber \\&= \eta (0,t_1]\cdot \eta (0,t_2] \cdots \eta (0,t_n]\cdot \frac{1}{n!}e^{-\eta (t)} \nonumber \\&= \frac{\eta (0,t_1]}{\eta (0,t_2]}\cdot \left(\frac{\eta (0,t_2]}{\eta (0,t_3]}\right)^2 \cdot \left(\frac{\eta (0,t_3]}{\eta (0,t_4]}\right)^3 \cdots \left(\frac{\eta (0,t_n]}{\eta (t)}\right)^n \cdot \frac{\eta (t)^n}{n!} e^{-\eta (t)}. \) 

For our purpose it suffices to show that
\(\frac{\eta (0,t_1]}{\eta (0,t_2]},\, \frac{\eta (0,t_2]}{\eta (0,t_3]},\,\cdots , \frac{\eta (0,t_n]}{\eta (t)},\,\eta (t)\) 

are totally independent, and
\(\frac{\eta (0,t_{k}]}{\eta (0,t_{k+1}]}\sim \mathrm {Beta}(\gamma t_k,\gamma (t_{k+1}-t_k)),\quad 1\le k\le n,\quad t_{n+1}:=t.\) 

The latter result follows from the property of Gamma r.v.'s.
For the former result,
we use the induction and assume the relation:
\(``\frac{\eta (0,t_1]}{\eta (0,t_2]}, \cdots ,\frac{\eta (0,t_{\ell }]}{\eta (0,t_{\ell +1}]},\, \eta (0,t_{\ell +1}] \quad \text{are totally independent''}\) 

holds for \(\ell =k-1\)  and show that it holds also for \(\ell =k\) .
Due to the property of Gamma r.v.'s
\(\frac{\eta (0,t_{k}]}{\eta (0,t_{k+1}]}\quad \text{and}\quad \eta (0,t_{k+1}]\quad \text{areindependent for all}\ \ell \le n,\) 

and the induction hypothesis with \(\ell =1\)  holds obviously.
Since \(\eta \)  is a Lévy process \(\eta (s,t]\)  is independent of the
filtration \(\mathcal {F}_s\)  for all \(0\le s<t<\infty \) .
Then
\(\frac{\eta (0,t_1]}{\eta (0,t_2]}, \cdots ,\frac{\eta (0,t_{k-1}]}{\eta (0,t_k]},\, \eta (0,t_k],\,\eta (t_{k},t_{k+1}]\quad \text{are totally independent}.\) 

Since a random set \((\eta (0,t_k]/\eta (0,t_{k+1}],\,\eta (0,t_{k+1}])\)  in
(REF ) is
included in \(\sigma \) -field by \(\lbrace \eta (0,t_k],\,\eta (t_k,t_{k+1}]\rbrace \) , it
is independent of \(\eta (0,t_{k-1}]/\eta (0,t_k]\)  by
(REF ) together with e.g. [1]}.
Now keeping (REF ) in mind, we apply the
relation between pairwise and total independence ([2]}) from the right-hand side of
(REF ) with \(\ell =k\) . This yields the desired total independence for \(\ell =k\) .
